Effect of Orientation Selection Scheme of Nucleus on Discontinuous Dynamic Recrystallization: Analysis with Multiscale Modeling Approach

Abstract

Effect of crystallographic orientation of nucleus on discontinuous dynamic recrystallization (DDRX) behavior in 304LN stainless steel is investigated using a multiscale model, namely a coupled crystal plasticity finite element method and DDRX-based cellular automata model. The three orientation selection schemes of nucleus are specially exploited in the simulation; i.e., (1) random orientation, (2) inheritance of orientation of parent deformed grain, and (3) generalized strain energy release maximization theory. The DDRX behaviors such as flow stress, DDRX volume fraction, grain size, and texture predicted by the three schemes are compared and the differences are explained through the simulated microstructure evolutions. This study suggests that it is reasonable to assign a random orientation to the nucleus through comparisons with experimental evidence.

1. Introduction

Discontinuous dynamic recrystallization (DDRX) is a metallurgical phenomenon commonly observed when metallic materials with low-to-medium stacking fault energy are subject to high temperature deformation.¹⁾ In terms of mechanical behavior, the DDRX phenomenon shows that peak appears in the flow stress of initial hardening stage, followed by the stress softening through recrystallization. It is well known that this phenomenon is caused by the multiple occurrences of nucleation and grain growth due to the accumulated stored energy under deformation at high temperature, which results in a dislocation-free state in a microstructure. Therefore, the DDRX causes grain refinement that allows beneficial combinations of strength and ductility by the multiple recrystallization process. However, there have been controversies on the crystallographic orientation selection for the newly formed nucleus.²⁾ Texture in metallic materials is closely related to mechanical anisotropy and response.³⁾ Therefore, it is essential to understand the mechanism of nucleation and its effect on the DDRX behavior.

Recently, various numerical methods have been proposed to simulate the DDRX behavior. The DDRX theory has been combined with meso-scale computational models such as Monte Carlo,⁴⁾ cellular automata (CA),⁵⁾ and phase field⁶⁾ models to simulate microstructure evolution by DDRX. Among them, the CA model has gain attentions due to its straightforward time and length scale calibrations and low computational costs.⁷⁾ However, the individual CA model has limitations in predicting heterogeneous deformation considering the grain orientations. Therefore, the CA models have been coupled with the crystal plasticity finite element method (CPFEM) to consider the influence of orientation on DDRX behavior.^8,9) Even in these modeling studies, there was no clear basis for orientation selection during the nucleation process, thus lacking robust physical validity. For example, Li et al.⁸⁾ inherited the orientation of the deformed matrix to the nucleus, and Tang et al.⁹⁾ assigned a random orientation to the nucleus. Therefore, for accurate DDRX modeling, it is necessary to investigate the influence of orientation selection during nucleation on DDRX behavior and to present an appropriate orientation selection scheme.

In this study, the effect of the crystallographic orientation assigned to the newly formed nucleus on DDRX behavior is studied through a multiscale model in the grain level. In our recent work, a fully coupled crystal plasticity-cellular automata model was developed to predict the heterogeneous deformation and DDRX behavior of 304LN stainless steel, which were successfully validated by the experimental.¹⁰⁾ Therefore, by extending the multiscale model, this study further examines the variation of DDRX behavior such as flow stress, grain size evolution, DDRX kinetics, and DDRX texture with respect to various orientation selection approaches.

2. Multiscale Modeling of DDRX

2.1 Crystal plasticity

The elastoplastic deformation kinematics of a single metal crystal is explained by decomposing the deformation gradient tensor into elastic and plastic deformation gradients, and the plastic deformation gradient evolves by the plastic shear strain rate on slip systems.¹¹⁾   

\begin{equation} \mathbf{F} = \mathbf{F}^{e}\mathbf{F}^{p} \end{equation} (1)

  

\begin{equation} \mathbf{L}^{p} = \dot{\mathbf{F}}^{p}\mathbf{F}^{p^{-1}} = \sum_{\alpha}\dot{\gamma}^{\alpha}(\boldsymbol{d}_{0}^{\alpha} \otimes \boldsymbol{n}_{0}^{\alpha}) \end{equation} (2)

where F^e and F^p are the elastic and plastic deformation gradients, respectively, L^p is the plastic velocity gradient, $\dot{\gamma }^{\alpha }$ is the plastic shear strain rate, $\boldsymbol{d}_{0}^{\alpha }$ and $\boldsymbol{n}_{0}^{\alpha }$ are the time-independent slip direction and slip plane normal of slip system α, respectively. The plastic shear strain rate is calculated using the phenomenological flow rule as follows.   

\begin{equation} \dot{\gamma}^{\alpha} = \dot{\gamma}_{0}\left|\frac{\tau^{\alpha}}{\tau_{cr}^{\alpha}}\right|^{n}\mathop{\text{sgn}}\nolimits (\tau^{\alpha}) \end{equation} (3)

with   

\begin{equation} \tau^{\alpha} = \mathbf{S}:(\boldsymbol{d}_{0}^{\alpha} \otimes \boldsymbol{n}_{0}^{\alpha})\ \text{and}\ \tau_{cr}^{\alpha} = \tau_{0}^{\alpha} + \tau_{\rho}^{\alpha} + \tau_{\textit{HP}}^{\alpha} \end{equation} (4)

where $\dot{\gamma }_{0}$ is the reference shear strain rate, n is the inverse strain rate sensitivity parameter, τ^α and $\tau_{cr}^{\alpha }$ are the resolved shear stress and critical resolved shear stress on slip system α, and S denotes the second Piola-Kirchhoff stress. $\tau_{0}^{\alpha }$, $\tau_{\rho }^{\alpha }$, and $\tau_{\textit{HP}}^{\alpha }$ are the initial slip resistance, inter-dislocation resistance, and resistance encountered by barrier effect due to grain boundaries, respectively, and are expressed as follows.   

\begin{equation} \tau_{0}^{\alpha} = C_{0}\dot{\varepsilon}^{n_{0}}\exp \left(-\frac{T}{T_{0}}\right) \end{equation} (5)

  

\begin{equation} \tau_{\rho}^{\alpha} = \chi\mu b\sqrt{\sum_{\beta = 1}^{N_{s}}\mathbf{h}_{\alpha\beta}\rho^{\beta}} \end{equation} (6)

  

\begin{equation} \tau_{\textit{HP}}^{\alpha} = \frac{H\mu\sqrt{b}}{\sqrt{d}} \end{equation} (7)

where T is deformation temperature, $\dot{\varepsilon }$ is the strain rate, χ is the inter-dislocation parameter, μ is the shear modulus, b is the magnitude of the Burgers vector, N_s is the total number of slip systems, h_αβ is the interaction matrix which is defined as n^α · l^β, where l^β is the dislocation line vector for slip system β. ρ^β is the dislocation density on slip system β, and d the diameter of a grain. And C₀, n₀, T₀, and H are model constants. Here, the dislocation density of the α-th slip system is evolved by the following Kocks-Mecking equation.¹²⁾   

\begin{align} &\dot{\rho}^{\alpha} = \left(k_{1}\sqrt{\sum_{\beta = 1}^{N_{s}}\rho^{\beta}} {}- k_{2}\rho^{\alpha}\right) \cdot |\dot{\gamma}^{\alpha}|;\\ & k_{2}(\dot{\varepsilon},T) = k_{20}\left[\dot{\varepsilon}\exp \left(\frac{Q_{\text{act}}}{RT}\right)\right]^{-\eta} \end{align} (8)

where k₁ represents the multiplication of dislocations, and k₂ represents the rate coefficient for dynamic recovery which is a function of strain rate and temperature. Q_act is the deformation activation energy, R is the ideal gas constant, k₂₀ and η are the material constants.

The Green-Lagrangian strain, which is a conjugated to the S, is related to the elastic deformation gradient, defined as,   

\begin{equation} \mathbf{E}^{e} = \frac{1}{2}(\mathbf{F}^{e^{T}}\mathbf{F}^{e} - \mathbf{I}) \end{equation} (9)

where I is the second-order identity tensor. Finally, the constitutive equation for stress response in a single crystal is governed by the linear elasticity law as follows.   

\begin{equation} \mathbf{S} = \mathbf{C}^{e}:\mathbf{E}^{e};\quad {\boldsymbol{\sigma}} = \det(\mathbf{F}^{e})^{-1}\mathbf{F}^{e}\mathbf{SF}^{e^{T}} \end{equation} (10)

where C^e is the elastic stiffness and $\boldsymbol{\sigma}$ is the Cauchy stress tensor.

2.2 DDRX-based CA model

The DDRX is initiated when the dislocation density nearby the grain boundary reaches the critical dislocation density during high temperature deformation, leading to nucleation and grain growth. It is well known that the critical dislocation density generally depends on the deformation temperature and strain rate.¹³⁾ Therefore, in this study, the following equation is adopted for the critical dislocation density.   

\begin{equation} \rho_{cr} = C_{\rho}\dot{\varepsilon}^{n_{\rho}}\exp \left(\frac{T_{\rho}}{T}\right) \end{equation} (11)

where C_ρ, n_ρ, and T_ρ are model parameters. And the nucleation rate per unit area also depends on the temperature and strain rate according to Ding and Guo,⁵⁾ expressed as   

\begin{equation} \dot{n} = C_{n}\dot{\varepsilon}^{n_{n}}\exp \left(\frac{T_{n}}{T}\right) \end{equation} (12)

where C_n, n_n, and T_n are model parameters. Then, the number of nuclei ΔN_NCL during the time step Δt can be expressed as,   

\begin{equation} \Delta N_{\textit{NCL}} = \dot{n}A_{\textit{GB}}\Delta t \end{equation} (13)

where A_GB is the total grain boundary area which is defined as A_GB = l₀²N_GB, where l₀ and N_GB are the cell size and the number of cells lying on the grain boundaries, respectively. When the i-th cell located along grain boundaries with ρ_i ≥ ρ_cr is selected, its nucleation probability (P_N) is given by   

\begin{equation} P_{N} = \frac{\rho_{i}}{\rho_{\max}} \end{equation} (14)

where ρ_max is the maximum dislocation density in the current time step. Therefore, cells with high stored energy are more likely to switch to new nucleus.

The growth velocity of newly formed nuclei (v) is governed by grain boundary mobility (M) and driving force (P) as follows.   

\begin{equation} v = MP \end{equation} (15)

with   

\begin{align} &M = \lambda \frac{\delta D_{0b}}{kT}\exp \left(-\frac{Q_{b}}{RT}\right)\left[1 - \exp \left\{-5\left(\frac{\theta}{\theta_{\textit{HAGB}}}\right)^{4}\right\}\right]\\ & \text{and}\ P = \frac{1}{2}\mu b^{2}(\rho_{m} - \rho_{\textit{DRX}}) - 2\frac{\gamma}{r_{\textit{eq}}} \end{align} (16)

where k is the Boltzmann constant, Q_b is the boundary diffusion activation energy, δ is the characteristic grain boundary thickness, D_0b is the boundary self-diffusion coefficient. And λ is coefficient of solute drag effect, which is a function of temperature and strain rate expressed as $\lambda = C_{\lambda }\dot{\varepsilon }^{n_{\lambda }}\exp ( - \frac{T_{\lambda }}{T} )$ where C_λ, n_λ, and T_λ are model fitting parameters. θ_HAGB is the misorientation angle for a high angle grain boundary (= 15°), and θ is the grain boundary misorientation, ρ_DRX and ρ_m are the dislocation densities of DDRX grains and its surrounding deformed matrix, respectively, r_eq is the equivalent radius of the DDRX grain, and γ is the grain boundary energy, given by   

\begin{equation} \gamma = \begin{cases} \dfrac{\mu b\theta_{\textit{HAGB}}}{4\pi(1 - \nu)}\dfrac{\theta}{\theta_{\textit{HAGB}}}\left[1 - \ln \left(\dfrac{\theta}{\theta_{\textit{HAGB}}}\right)\right] \\ \qquad\qquad \qquad\quad \text{for $0^{\circ} < \theta < 15^{\circ}$}\\ \dfrac{\mu b\theta_{\textit{HAGB}}}{4\pi(1 - \nu)} \qquad\quad \text{for $\theta \geq 15^{\circ}$} \end{cases} \end{equation} (17)

where ν is the Poisson’s ratio.

In the context of CA module, growth area of nucleus is considered to simulate grain growth after nucleation. In order for the deformed cell j to be switched to the DDRX state by the neighboring DDRX cells, the growth area in cell j by the neighboring DDRX cells (A_growth,j) must reach the cell area.   

\begin{equation} A_{\textit{growth},j} = \sum_{i = 1}^{N_{n}}\int_{t_{\textit{nucleation}}}^{t}\frac{A_{0}}{l_{ij}}v_{ij}dt;\quad A_{\textit{growth},j} \geq A_{0} \end{equation} (18)

where t and t_nucleation are the current time and the time at the moment of nucleation, v_ij is grain growth velocity of the DDRX neighbor cell i toward the deformed cell j, l_ij is the distance between cell i and j, and N_n is the total number of the neighbor cells. In this study, the Moore’s neighboring rule is adopted so that N_n is set to 8. Since the distance between cells changes due to deformation, the equiaxed grain growth is simulated taking this into account. After growth area reaches the cell size, its growth probability (P_G) is defined by,   

\begin{equation} P_{G} = \frac{N_{i}}{N_{n}} \end{equation} (19)

while N_i is the number of neighbor cells in DDRX state. In other words, the more it is surrounded by DDRX grains, the easier it grows. That is, the grain growth occurs easily if the deformed cell is surrounded by more DDRX cells.

2.3 Integration scheme of DDRX-based CACPFEM model

In order to integrate the DDRX-based CA model and CPFEM, the state variables in the two models should be shared and updated in real time to reflect the influences on the mechanical behavior of each model. The multiscale modeling scheme is illustrated in Fig. 1 and the descriptions for DDRX-based CACPFEM model are as follows.

Fig. 1

Multiscale modeling for simulating DDRX behavior.

Experimental studies for DDRX have confirmed that geometrically necessary dislocations within DDRX grains are significantly reduced, which facilitates the movement of statistically stored dislocations. In other words, the inter-dislocation interaction is reduced. Therefore, the coefficient of the inter-dislocation resistance of the deformed and the DDRX grain in the CPFEM can be set differently as follows.   

\begin{equation} \tau_{\rho}^{\alpha} = \begin{cases} \chi_{m}\mu b\sqrt{\displaystyle\sum_{\beta = 1}^{N_{s}}\mathbf{h}_{\alpha\beta}\rho_{\beta}} & \text{for deformed grain}\\ \chi_{\textit{DRX}}\mu b\sqrt{\displaystyle\sum_{\beta = 1}^{N_{s}}\mathbf{h}_{\alpha\beta}\tilde{\rho}_{\beta}} & \text{for DRX grain} \end{cases} \end{equation} (20)

where χ_m and χ_DRX denote the dislocation interaction coefficient of the deformed and DDRX grain, respectively, and the tilde denotes the state variable of a DDRX grain. Since a newly formed nucleus is dislocation-free state, the dislocation density and the deformation gradient of the cell that switched to the nucleus are initialized. And new orientation is given to the nucleus. In the grain growth part, the dislocation density and plastic deformation gradient are initialized and elastic deformation gradient and orientation are inherited by neighboring cell.   

\begin{equation} \text{Nucleation}{:}\ (\tilde{\mathbf{F}}^{e},\tilde{\mathbf{F}}^{p},\tilde{\rho}^{\alpha},\tilde{{\boldsymbol{\varphi}}}) \leftarrow (\mathbf{I},\mathbf{I},\rho_{\textit{low}}^{\alpha},\text{new orientation}) \end{equation} (21)

  

\begin{equation} \text{Grain growth}{:}\ (\tilde{\mathbf{F}}^{e},\tilde{\mathbf{F}}^{p},\tilde{\rho}^{\alpha},\tilde{{\boldsymbol{\varphi}}}) \leftarrow (\tilde{\mathbf{F}}^{e,\textit{neighbor}},\mathbf{I},\rho_{\textit{low}}^{\alpha},\tilde{{\boldsymbol{\varphi}}}^{\textit{neighbor}}) \end{equation} (22)

where $\boldsymbol{\varphi}$ is the crystallographic orientation and $\rho_{low}^{\alpha }$ is the initialized dislocation density. To initialize the deformation gradient, the relative deformation gradient (F^r) is introduced.   

\begin{equation} \mathbf{F} = \tilde{\mathbf{F}}\mathbf{F}^{r} \end{equation} (23)

In this study, the effect of orientation selection of newly formed nuclei on DDRX behavior is analyzed. Therefore, new orientation given to the nucleus is determined from the following three methods.

(1) Random orientation – ‘Case R’

The DDRX texture is experimentally known as a random texture, and a random orientation has been adopted in many studies on DDRX modeling.^14,15) Note that a random orientation given to the nucleus maintains a HAGB with the surrounding grains.

(2) Inheritance of orientation of the deformed grain – ‘Case I’

Several studies have shown that the DDRX grains inherit the orientation of the parent deformed grain.⁸⁾ This study also adopted it as one of the orientation selection methods.

(3) Generalized strain energy release maximization theory (GSERM) – ‘Case G’

The generalized strain energy release maximization theory³⁾ suggests that the orientation of the nucleus is determined to maximize the release of stored elastic energy. That is, the orientation of a DDRX grain is determined so that the absolute maximum stress direction (AMSD) of the deformed grain is parallel to the minimum Young’s modulus direction (MYMD) of a DDRX grain as follows.   

\begin{align} \Delta E_{\textit{release}} &= \int ({\boldsymbol{\sigma}}_{m} - {\boldsymbol{\sigma}}_{\textit{DRX}}):d{\boldsymbol{\varepsilon}} \\ &= \int ({\boldsymbol{\sigma}}_{m} - \mathbf{C}_{\textit{DRX}}^{e}:{\boldsymbol{\varepsilon}}):d{\boldsymbol{\varepsilon}} \end{align} (24)

where ΔE_release is the released strain energy and $\boldsymbol{\varepsilon}$ is the infinitesimal strain tensor. In this theory, it is assumed that AMSD is approximated as the effective slip direction (ESD), defined as   

\begin{equation} \bar{\boldsymbol{d}} = \sum_{\alpha = 1}^{N_{s}}\rho^{\alpha}\boldsymbol{d}_{0}^{\alpha} \end{equation} (25)

where $\bar{\boldsymbol{d}}$ is the ESD. Once the ESD is determined, the non-active slip direction orthogonal to the ESD is designated as the rotation axis. And the amount of rotation for parallelism with MYMD is calculated from the cross-product of the two vectors. Meanwhile, if there is no non-active slip direction orthogonal to the ESD, the orientation of the nucleus inherits the orientation of the deformed matrix.

The computational simulation procedure is shown in Fig. 2. The model is implemented in Abaqus/Standard with UMAT. In the model, a 200 × 200 square mesh of 2.5D RVE is used and the cell size is 1 µm; thus, the dimensions of 2.5D RVE model are 200 µm × 200 µm. The initial microstructure is generated using the DREAM.3D software¹⁶⁾ based on the experimental data. Note that the experimental data refer to Rout et al.¹⁴⁾ The initial microstructure information is presented in Fig. 3. The periodic boundary conditions are used. The model parameters of the DDRX-based CACPFEM model are referred to Park et al.¹⁰⁾ and fitted based on Case R.

Fig. 2

Schematics of the numerical procedure of DDRX-based CACPFEM model.

Fig. 3

Representative volume element for DDRX simulation. The color code represents along with the ND.

3. Results and Discussion

3.1 Flow stress

Figure 4 shows the flow stress predicted by the model with the three orientation selection schemes. Note that, since the model parameters were calibrated with Case R, it can be seen that Case R matches well with the experimental data. From the predicted results, the stress decreases as the temperature increases or the strain rate decreases. Also, in all cases, stress softening occurs with the same tendency as the experimental results. However, the mechanical response is different depending on the orientation of the nucleus. For all deformation conditions, the strain at peak stress is earliest in Case R and latest in Case I. Accordingly, the flow stress also increases in the order of Case R, Case G, and Case I. Here, since the same initial microstructure was used for all cases, the initial hardening stage and the initiation of DDRX is identical. Therefore, the difference in mechanical response is caused by the DDRX behavior.

Fig. 4

Comparison of the predicted flow stress between Case R, Case I, and Case G: (a) 1 s⁻¹; (b) 0.1 s⁻¹; (c) 0.01 s⁻¹.

3.2 DDRX volume fraction

Figure 5 compares the change of the DDRX volume fraction with respect to strain for all cases, which shows the commonly known Avrami-type kinetics.¹⁷⁾ The DDRX volume fraction becomes large as the temperature increased or the strain rate decreased. In terms of the orientation section approaches, the DDRX kinetics are fastest in Case R and the slowest in Case I. Therefore, the flow stress in Fig. 4 is different due to the difference in the DDRX kinetics in each case.

Fig. 5

Comparison of the predicted DDRX volume fraction between Case R, Case I, and Case G: (a) 1 s⁻¹; (b) 0.1 s⁻¹; (c) 0.01 s⁻¹.

3.3 Grain size evolution

The advantages of DDRX process is that it induces grain refinement resulting in superior strength and ductility. The predicted grain sizes by the model decrease significantly compared to the initial grain size. Also, the grain size increases as the temperature increases or the strain rate decreases. That is, increasing grain boundary mobility at higher temperature accelerates the grain growth velocity and the lower strain rate provides sufficient time for the grain boundary movement, resulting in a larger grain size. And, the grain size is large in the order of Case R, Case G, and Case I. It seems that there are differences in grain growth depending on the crystallographic orientation during the DDRX process.

3.4 Microstructure evolution

For a more in-depth investigation of the effect of orientation imparted to a nucleus, the microstructure evolution by DDRX in specific region A and B marked with white dashed boxes in Fig. 3 was examined under the deformation condition of 1000°C and 0.1 s⁻¹. Figure 7 shows the evolution of microstructure and dislocation density distribution with increasing strain right after nucleation in the region A. It can be seen that the deformation occurs non-homogeneously depending on the grain, and in particular, the dislocation density is concentrated around grain boundary. Also, the dislocation density of the nucleus is initialized to small value, and grain growth occurs due to the difference in stored energy with the deformed matrix. Dislocations multiply again inside the DDRX grains due to the deformation.

In Case R in Fig. 7(a), the nucleus receives a random orientation satisfying HAGB with the surrounding matrix, resulting in the equiaxed DDRX grain. Meanwhile, since the orientation of the parent deformed grain is transferred to a nucleus in Case I, the nucleus forms LAGB with the parent matrix in Fig. 7(b). Here, a slight misorientation angle exists as a texture gradient due to heterogeneous deformation. Equation (16) describes the dependence of the grain boundary mobility on the misorientation, leading that the smaller the orientation difference angle, the sharply lower the grain boundary mobility. Therefore, it is confirmed that the growth of the nucleus to the parent matrix with a small misorientation angle is very slow. Li et al.⁸⁾ showed faster grain growth as the misorientation angle was smaller, but their work did not consider the misorientation angle-dependence of grain boundary mobility. Also, in Case G, the nucleus received the orientation of the deformed grain without any special orientation selection. Note that, according to the GSERM theory, if there is no non-active slip direction perpendicular to the ESD, the orientation of the deformed matrix is inherited. Therefore, DDRX grain growth in Case I and Case G is slower than in Case R in the A region.

Figure 8 shows the evolution of microstructure and dislocation density distribution in the region B shown in Fig. 3. Case R and Case I show equiaxed and non-equiaxed growth of the generated nucleus same as in the region A. In the Case G, the new nucleus was given a new orientation by the GSERM theory, and this also constitutes HAGB with the surrounding deformed grains, resulting in the equiaxed DDRX grain. It is confirmed that the dislocation density near the grain boundary in Case I was accumulated more slowly than that in Case R. For this reason, the difference in the stored energy between the deformed grains and the DDRX grains at the grain boundary is larger in Case I than that in Case R. This corresponds to the faster grain growth in Case I than Case R.

In summary, in Case R, the generated nucleus always grows at a similar rate in all directions to form equiaxed DDRX grains, i.e. high DDRX kinetics. On the other hand, Case I shows low DDRX kinetics because of irregular grain growth. And the orientation is determined according to the slip activity and shows intermediate kinetics in Case G. That is, the DDRX kinetics are greater in the following order: Case R > Case G > Case I. Therefore, as shown in Fig. 6, the grain size consistently shows Case R > Case G > Case I under all deformation conditions because of the difference in grain growth, and the DDRX volume fraction in Fig. 5 is in the same vein. Also, due to the difference in DDRX kinetics, the flow stress level follows Case R < Case G < Case I in Fig. 4.

Fig. 6

Comparison of the predicted grain size evolution between Case R, Case I, and Case G: (a) 1 s⁻¹; (b) 0.1 s⁻¹; (c) 0.01 s⁻¹.

Fig. 7

Evolution of microstructure and dislocation density distribution with strain under the deformation condition of 1000°C and 0.1 s⁻¹ in region A marked in Fig. 3: (a) Case R, (b) Case I, (c) Case G.

Fig. 8

Evolution of microstructure and dislocation density distribution with strain under the deformation condition of 1000°C and 0.1 s⁻¹ in region B marked in Fig. 3: (a) Case R, (b) Case I, (c) Case G.

3.5 DDRX texture

Figure 9 shows comparison between the predicted textures and experimental results at a true strain of 0.7 under the deformation condition of 1000°C and 0.1 s⁻¹. The deformation texture under compression test of the FCC structure at room temperature is a ⟨110⟩ fiber, which means that grains are rotated such that a (110) plane is perpendicular to the compression axis. However, the experimental observations in Fig. 9(a) shows that the DDRX texture is randomized by DDRX. The results of Case R are random texture consistent with the experimental result. On the other hand, the intensity of the ⟨110⟩ fiber in Case I is strong since a nucleus are assigned to the deformed orientation. In Case G, initial orientation of a nucleus may be set to the deformed orientation as shown in Fig. 7, moreover, random orientation is also given to a nucleus as shown in Fig. 8, leading that the intensity range of the pole figure of Case G is narrower than that of Case I. Therefore, in DDRX modeling, it can be concluded that randomly assigning the orientation of newly generated nuclei is most consistent with the experiment in terms of crystallography.

Fig. 9

Comparison of DDRX textures at strain of 0.7 under 1000°C and 0.01 s⁻¹ between (a) experimental results, (b) Case R, (c) Case I, (d) Case G.

4. Conclusion

In this study, effect of selection scheme of crystallographic orientation of nucleus on the DDRX behavior in 304LN stainless steel was investigated by using a multiscale model. The computational approach couples the crystal plasticity finite element method and the DDRX-based cellular automata model. The DDRX behaviors were predicted by introducing three different orientation selection methods: random orientation, inheritance of orientation of parent deformed grain, and the GSERM theory. Then, the results predicted by the three approaches were compared with each other to confirm the effect on the DDRX behavior. The DDRX behaviors including flow stress, DDRX volume fraction, and grain size evolution were all different from each other, which is attributed to differences in deformation behavior and grain growth due to orientation given to a nucleus. In addition, it was concluded that it is most reasonable to assign a random orientation to a nucleus by comparing the DDRX texture predicted by each approach with the experimental observation.

Acknowledgments

This research was supported by Korea Institute for Advancement of Technology (KIAT) grant funded by the Korea Government (MOTIE) (P0002019, Human Resource Development Program for Industrial Innovation). Also, MGL appreciates partial supports from National Research Foundation (NRF) of Korea (Grant No. 2022R1A2C2009315) and Institute of Engineering Research at Seoul National University. HNH appreciates the support from NRF Grant funded by the Korean Government (MSIP) (No. NRF-2021M3H4A6A01045764).

REFERENCES

• 1)   Y.  Cai,  S.  Xiang and  Y.  Tan: Met. Mater. Int. (2022) 1–14. doi:10.1007/s12540-021-01139-8
• 2)   K.  Huang and  R.E.  Logé: Mater. Des. 111 (2016) 548–574. doi:10.1016/j.matdes.2016.09.012
• 3)   K.M.  Min,  W.  Jeong,  S.H.  Hong,  C.A.  Lee,  P.R.  Cha,  H.N.  Han and  M.G.  Lee: Int. J. Plast. 127 (2020) 102644. doi:10.1016/j.ijplas.2019.102644
• 4)   P.  Peczak and  M.J.  Luton: Acta Metall. Mater. 41 (1993) 59–71. doi:10.1016/0956-7151(93)90339-T
• 5)   R.  Ding and  Z.X.  Guo: Acta Mater. 49 (2001) 3163–3175. doi:10.1016/S1359-6454(01)00233-6
• 6)   Y.  Cai,  C.Y.  Sun,  Y.L.  Li,  S.Y.  Hu,  N.Y.  Zhu,  E.I.  Barker and  L.Y.  Qian: Int. J. Plast. 133 (2020) 102773. doi:10.1016/j.ijplas.2020.102773
• 7)   H.W.  Lee and  Y.-T.  Im: Mater. Trans. 51 (2010) 1614–1620. doi:10.2320/matertrans.M2010116
• 8)   H.  Li,  X.  Sun and  H.  Yang: Int. J. Plast. 87 (2016) 154–180. doi:10.1016/j.ijplas.2016.09.008
• 9)   T.  Tang,  G.  Zhou,  Z.  Li,  D.  Li,  L.  Peng,  Y.  Peng,  P.  Wu,  H.  Wang and  M.G.  Lee: Int. J. Plast. 116 (2019) 159–191. doi:10.1016/j.ijplas.2019.01.001
• 10)   J.  Park,  M.  Rout,  K.M.  Min,  S.F.  Chen and  M.G.  Lee: Mech. Mater. 167 (2022) 104248. doi:10.1016/j.mechmat.2022.104248
• 11)   S.R.  Kalidindi,  C.A.  Bronkhorst and  L.  Anand: J. Mechan. Phys. Solids 40 (1992) 537–569. doi:10.1016/0022-5096(92)80003-9
• 12)   H.  Mecking and  U.F.  Kocks: Acta Metall. 29 (1981) 1865–1875. doi:10.1016/0001-6160(81)90112-7
• 13)   J.  Yu,  I.Y.  Moon,  H.W.  Jeong,  H.W.  Lee,  J.H.  Kim and  S.H.  Kang: Met. Mater. Int. (2022) 1–17. doi:10.1007/s12540-022-01194-9
• 14)   M.  Rout,  S.  Biswas,  R.  Ranjan,  S.K.  Pal and  S.B.  Singh: Metall. Mater. Trans. A 49 (2018) 864–880. doi:10.1007/s11661-017-4447-5
• 15)   G.  Zhou,  Z.  Li,  D.  Li,  Y.  Peng,  H.S.  Zurob and  P.  Wu: Int. J. Plast. 91 (2017) 48–76. doi:10.1016/j.ijplas.2017.01.001
• 16)   M.A.  Groeber and  M.A.  Jackson: Integr. Mater. Manuf. Innov. 3 (2014) 56–72. doi:10.1186/2193-9772-3-5
• 17)   J.J.  Jonas,  X.  Quelennec,  L.  Jiang and  É.  Martin: Acta Mater. 57 (2009) 2748–2756. doi:10.1016/j.actamat.2009.02.033